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arXiv:2505.13405

A Dataless Reinforcement Learning Approach to Rounding Hyperplane Optimization for Max-Cut

Gabriel Maliakal, Ismail Alkhouri, Alvaro Velasquez, Adam Michael Alessio, Saiprasad Ravishankar

Published May 19, 2025• 0 citations
arXivHTMLPDF

Abstract

The Maximum Cut (MaxCut) problem is NP-Complete, and obtaining its optimal solution is NP-hard in the worst case. As a result, heuristic-based algorithms are commonly used, though their design often requires significant domain expertise. More recently, learning-based methods trained on large (un)labeled datasets have been proposed; however, these approaches often struggle with generalizability and scalability. A well-known approximation algorithm for MaxCut is the Goemans-Williamson (GW) algorithm, which relaxes the Quadratic Unconstrained Binary Optimization (QUBO) formulation into a semidefinite program (SDP). The GW algorithm then applies hyperplane rounding by uniformly sampling a random hyperplane to convert the SDP solution into binary node assignments. In this paper, we propose a training-data-free approach based on a non-episodic reinforcement learning formulation, in which an agent learns to select improved rounding hyperplanes that yield better cuts than those produced by the GW algorithm. By optimizing over a Markov Decision Process (MDP), our method consistently achieves better cuts across large-scale graphs with varying densities and degree distributions.

DOI
10.48550/arxiv.2505.13405

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INFORMS journal on computing•Jul 16, 2025•0 citations
The Maximum Cut (MaxCut) problem is NP-Complete, and obtaining its optimal solution is NP-hard in the worst case. As a result, heuristic-based algorithms are commonly used, though their design often requires significant domain expertise. More recently, learning-based methods trained on large (un)labeled datasets have been proposed; however, these approaches often struggle with generalizability and scalability. A well-known approximation algorithm for MaxCut is the Goemans-Williamson (GW) algorithm, which relaxes the Quadratic Unconstrained Binary Optimization (QUBO) formulation into a semidefinite program (SDP). The GW algorithm then applies hyperplane rounding by uniformly sampling a random hyperplane to convert the SDP solution into binary node assignments. In this paper, we propose a training-data-free approach based on a non-episodic reinforcement learning formulation, in which an agent learns to select improved rounding hyperplanes that yield better cuts than those produced by the GW algorithm. By optimizing over a Markov Decision Process (MDP), our method consistently achieves better cuts across large-scale graphs with varying densities and degree distributions.
We present a novel approach to accelerate the Goemans-Williamson (GW) randomized rounding procedure for quadratic unconstrained binary optimization (QUBO) problems. Instead of solving the conventional semi-definite programming (SDP) relaxation, which is computationally expensive, we employ a difference-of-convex (DC) optimization framework to efficiently approximate the SDP solution. The DC optimization produces candidate vectors that are then used within the GW randomized rounding scheme to generate high-quality binary solutions. Furthermore, we perform direct expectation minimization over manifolds of matrices with limited rank to further enhance the solution quality. Our method is benchmarked on real-world QUBO instances, including inverse kinematics problems, and compared against state-of-the-art solvers, such as quantum-inspired algorithms, demonstrating competitive approximation guarantees alongside substantial computational gains.
This repository contains the source code, computational experiments, analysis, and benchmarks accompanying the paper A Direct SDP-Based Framework for Certified Bounds for Constrained Quadratic Binary Optimization Problems. The repository implements a direct semidefinite programming (SDP) approach for constrained quadratic binary optimization, designed to obtain strong bounds for exact optimization. The SDP relaxation is strengthened using quadratized linear constraints and triangle (hypermetric) inequalities. To address the numerical infeasibility of SDP solutions obtained from numerical solvers such as MOSEK, the framework includes a correction procedure that converts approximate SDP solutions into mathematically certified bounds suitable for use in Branch-and-Bound algorithms. The repository also implements an adaptation of the Goemans–Williamson randomized hyperplane rounding scheme to constrained binary optimization, together with problem-specific feasibility restoration. Computational experiments are provided for the Sparsest k-Subgraph and Maximum Independent Set problems. The work presented in this repository was funded and developed under the QBIQ project (project no. 644), funded by the Slovenian Research and Innovation Agency (ARIS) through the Applied Projects programme. The code and accompanying data are intended to facilitate reproducibility of the computational results and provide a foundation for further research on exact SDP-based methods for constrained quadratic binary optimization.
Quadratic Unconstrained Binary Optimization (QUBO) problems are prevalent in various applications and are known to be NP-hard. The seminal work of Goemans and Williamson introduced a semidefinite programming (SDP) relaxation for such problems, solvable in polynomial time that upper bounds the optimal value. Their approach also enables randomized rounding techniques to obtain feasible solutions with provable performance guarantees. In this work, we identify instances of QUBO problems where matrix multiplicative weight methods lead to quantum and quantum-inspired algorithms that approximate the Goemans-Williamson SDP exponentially faster than existing methods, achieving polylogarithmic time complexity relative to the problem dimension. This speedup is attainable under the assumption that the QUBO cost matrix is sparse when expressed as a linear combination of Pauli strings satisfying certain algebraic constraints, and leverages efficient quantum and classical simulation results for quantum Gibbs states. We demonstrate how to verify these conditions efficiently given the decomposition. Additionally, we explore heuristic methods for randomized rounding procedures and extract the energy of a feasible point of the QUBO in polylogarithmic time. While the practical relevance of instances where our methods excel remains to be fully established, we propose heuristic algorithms with broader applicability and identify Kronecker graphs as a promising class for applying our techniques. We conduct numerical experiments to benchmark our methods. Notably, by utilizing tensor network methods, we solve an SDP with D=250D = 2^{50}D=250 variables and extract a feasible point which is certifiably within 0.15%0.15\%0.15% of the optimum of the QUBO through our approach on a desktop, reaching dimensions millions of times larger than those handled by existing SDP or QUBO solvers, whether heuristic or rigorous.
The Quantum Approximate Optimization Algorithm (QAOA) is a quantum algorithm that finds approximate solutions to problems in combinatorial optimization, especially those that can be formulated as a Quadratic Unconstrained Binary Optimization (QUBO) problem. In prior work, researchers have considered various ways of "warm-starting" QAOA by constructing an initial quantum state using classically-obtained solutions or information; these warm-starts typically cause QAOA to yield better approximation ratios at much lower circuit depths. For the Max-Cut problem, one warm-start approaches constructs the initial state using the high-dimensional vectors that are output from an SDP relaxation of the corresponding Max-Cut problem. This work leverages these semidefinite warmstarts for a broader class of problem instances by using a standard reduction that transforms any QUBO instance into a Max-Cut instance. We empirically compare this approach to a "QUBO-relaxation" approach that relaxes the QUBO directly. Our results consider a variety of QUBO instances ranging from randomly generated QUBOs to QUBOs corresponding to specific problems such as the traveling salesman problem, maximum independent set, and portfolio optimization. We find that the best choice of warmstart approach is strongly dependent on the problem type.
Quadratic Unconstrained Binary Optimization (QUBO) is a versatile approach used to represent a wide range of NP-hard Combinatorial Optimization (CO) problems through binary variables. The transformation of QUBO to an Ising Hamiltonian is recognized as an effective method for solving key optimization problems using quantum algorithms. Recently, PI-GNN, a generic framework, has been proposed to address CO problems over graphs based on QUBO with Hamiltonian loss function to train the underlying GNN architecture. Though PI-GNN is highly scalable, it exhibits a noticeable decrease in terms of the number of satisfied constraints with higher graph densities. In this paper, firstly, we identify the limitations and empirically present our strategy to improve PI-GNN’s performance. Secondly, we formulate and evaluate two strategies to integrate QUBO-Hamiltonian as the generic loss function in Reinforcement learning-based (RL) frameworks. The major contribution of our work lies in understanding the feasibility and quality of the QUBO-based generic reward function in an unsupervised RL setup in addressing graph-based CO problems. Empirically, through our empirical evaluation (Our implementation can be found inhttps://tinyurl.com/5apnymz7), we have observed up to 44% improvement in terms of the number of satisfied constraints over PI-GNN in a representative Max-Cut problem.
The subgraph sampling scheme (SSS) is a technique originally introduced for Markov random fields. It is a powerful tool for designing heuristic algorithms for max-cut, quadratic unconstrained binary optimization (QUBO), and other optimization problems. The first application of SSS in combinatorial optimization, combined with dynamic programming, is in Selby’s heuristic. This algorithm is shown to outperform quantum annealing for solving max-cut problems on chimera graphs. Leveraging SSS, we introduce two new algorithms. One is designed to handle general graphs, whereas the other is specifically tailored for toroidal grid graphs. To assess the effectiveness of these algorithms, we conducted a comprehensive evaluation. We used the same methodology, test bed, and set of 37 well-established heuristics for max-cut and QUBO problems as described in a recent study of Dunning, Gupta, and Silberholz. Notably, all three SSS-based algorithms consistently achieve top rankings in terms of performance. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work was supported by the funding program Horizon 2020 - Excellent Science - Marie Skłodowska-Curie Actions of the European Commission (Grant MINOA- Mixed-Integer Non-Linear Optimization Applications/764759]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0812 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0812 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .